A group of retailers will buy 96 televisions from a wholesaler if the price is $500 and 136 if the price is $450. The wholesaler is willing to supply 84 if the price is $420 and 164 if the price is $510. Assuming the resulting supply and demand functions are linear, find the equilibrium point for the market.
To find the equilibrium point for the market, we need to determine the price and quantity at which the supply and demand are equal. We can do this by analyzing the given data and forming equations for the supply and demand functions.
Let's define the price as "P" and the quantity as "Q."
From the first set of data, we can form the demand equation:
When the price is $500, the demand is 96 televisions.
This gives us a point (500, 96) on the demand curve.
Similarly, from the second set of data, we can form another point on the demand curve: (450, 136).
Using these two points, we can calculate the slope of the demand curve:
slope = (change in quantity) / (change in price)
= (136 - 96) / (450 - 500)
= 40 / (-50)
= -0.8
Thus, the demand function becomes:
Q = -0.8P + b ----------(1)
Now, let's analyze the supply data.
Using the third set of data, we can form a point on the supply curve: (420, 84).
From the fourth set of data, we can form another point on the supply curve: (510, 164).
Calculating the slope of the supply curve:
slope = (164 - 84) / (510 - 420)
= 80 / 90
= 8/9
Therefore, the supply function becomes:
Q = (8/9)P + c ----------(2)
To determine the equilibrium point, we need to find the price and quantity at which the supply and demand functions intersect. We can do this by setting the demand and supply equations equal to each other:
-0.8P + b = (8/9)P + c
Combining like terms:
(8/9)P + 0.8P = c - b
Simplifying:
(80/90 + 8/9)P = c - b
(88/90)P = c - b
(8/9)P = c - b
Now, to find the specific values for P, c, and b, we use one of the given data points. Let's choose (420, 84) since it is a supply data point:
84 = (8/9)(420) + c
Solving for c:
c = 84 - (8/9)(420)
c ≈ 84 - 373.33
c ≈ -289.33
Now, we can substitute the value of c into the equation:
(8/9)P = (8/9)P - 289.33 - b
To solve for b, we need another data point. Let's use (500, 96), which is a demand data point:
96 = -0.8(500) + b
Solving for b:
b = 96 + (0.8)(500)
b = 96 + 400
b = 496
Finally, we can substitute the values of b and c back into the equation:
(8/9)P = (8/9)P - 289.33 - 496
Simplifying:
0 = -785.33
Since the equation leads to a contradiction, it means there is no equilibrium point using the given data. The supply and demand functions do not intersect within the range of prices and quantities provided.